Problem 66 - A Graphical Approach to College Algebra

Question

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}+y^{2} &=4 \\ 2 x^{2}-3 y^{2} &=-12 \end{aligned}$$

Step-by-Step Solution

Answer

The solutions are \((0, 2)\) and \((0, -2)\).

Step 1: Label the Equations

Let's label the given equations for clarity: Equation (1): \( x^2 + y^2 = 4 \)Equation (2): \( 2x^2 - 3y^2 = -12 \)

Step 2: Solve Equation (1) for x²

In Equation (1), solve for \( x^2 \) to express it in terms of \( y^2 \):\[ x^2 = 4 - y^2 \]

Step 3: Substitute into Equation (2)

Substitute \( x^2 = 4 - y^2 \) from Step 2 into Equation (2):\[ 2(4 - y^2) - 3y^2 = -12 \]

Step 4: Simplify the Equation

Distribute and combine like terms in the equation:\[ 8 - 2y^2 - 3y^2 = -12 \]Combine like terms:\[ 8 - 5y^2 = -12 \]

Step 5: Solve for y²

Isolate \( y^2 \) on one side:\[ 8 + 12 = 5y^2 \]\[ 20 = 5y^2 \]Divide by 5:\[ y^2 = 4 \]

Step 6: Find y Values

Take the square root of both sides to solve for \( y \):\( y = 2 \) or \( y = -2 \)

Step 7: Substitute y Values into x² Equation

Substitute \( y = 2 \) and \( y = -2 \) into \( x^2 = 4 - y^2 \):For \( y = 2 \):\[ x^2 = 4 - 2^2 = 0 \]So, \( x = 0 \).For \( y = -2 \):\[ x^2 = 4 - (-2)^2 = 0 \]So, \( x = 0 \).

Step 8: Final Solutions

All real solutions for the system are based on the values found:\( (x, y) = (0, 2) \) and \( (0, -2) \) are the solutions.

Key Concepts

Analytical Solution

Solving a system of nonlinear equations analytically involves finding the exact values of variables that satisfy all equations in the system. This method contrasts with numerical solutions, where approximate values are found using iterative techniques. To solve analytically:

• Identify each equation in the system clearly.
• Use algebraic manipulation to express variables in terms of others.
• Simplify the equations to isolate variables step by step.

In our exercise, we labeled the equations and used algebraic substitution to break down the equations into simpler expressions. This approach allows us to find exact values of the variables, which in this case, determines the real points at which these curves intersect.

Real Solutions

Real solutions are the set of values for variables that satisfy the equations within the real number system. With nonlinear systems, these solutions may represent geometric intersections, such as points where curves meet on a graph.

• Check each potential solution in the original equations to ensure correctness.
• Solutions must satisfy all equations simultaneously.

For our system, solving gives \( (x, y) = (0, 2) \) and \( (0, -2) \) as real solutions. This means each solution pair satisfies both equations exactly, providing tangible and accurate results. Testing solutions by substitution back into the original equations ensures their validity.

Substitution Method

The substitution method is a technique used to solve systems of equations, often effective for nonlinear types. It involves replacing one variable with an equivalent expression derived from another equation. This simplifies the system and makes it easier to solve.In our problem, we first solved for one variable in terms of others. For example:

• Solve for \( x^2 \) in terms of \( y^2 \) from the first equation.
• Substitute this expression into the second equation.

This method reduces the problem from two equations in two variables to a single equation in one variable, which we can solve more easily. In the exercise, it was crucial to solve \( y^2 \) first, which then allowed us to find the corresponding \( x \) values by substitution. This approach ensures all solutions are captured, making it a robust method for tackling nonlinear systems.